Fractals/Iterations in the complex plane/construct

How to construct a map with desired properities ?

Shape

Shape of

Julia set
critical orbit
external rays landind on the repelling cycle ( spirals)

Relation between shape types and dynamics:

n-th arm spiral: attracting or repelling n-periodic orbit ( cycle)
closed curve: Siegel disc ( rotation)
n-th arm star = period n parabolic root

Modelling or shaping

Usually one should controll 2 parameters:

fixed point
period p orbit

Julia set is modelled by critical orbit
Julia set is modelled by critical orbit
Siegel disk critical orbit is modelled by period 7 repelling orbit
Siegel disk critical orbit is modelled by period 2 repelling orbit
Julia set is modelled by repelling period 6 orbit and it's external rays

See

Shaping spirals

Conversions and evolutions

from Siegel to parabolic star
Superattracting through attracting spiral to parabolic star ( along internal ray)

Examples

Rational functions with prescribed critical points by I. Scherbak
constructing map with Fatou components of desired type^[1]
Constructing polynomials whose Julia set resemble a desired shape^[2]
Constructing polynomials that have an attracting cycle visiting predefined points in the plane ^[3]
constructuing QUARTIC JULIA SETS INCLUDING ANY TWO COPIES OF QUADRATIC JULIA SETS
constructing critically finite real polynomial maps with specified combinatorics^[4]

roots

The fundamental theorem of algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots^[5]^[6]

The factor theorem^[7] states that a polynomial $f(z)$ has a factor $(z-r)^{m}$ if and only if $f(r)=0$ (i.e. $r$ is a root of multiplicity^[8] m)

Examples

The polynomial^[9]

  $f(z)=(z-0)^{2}(z-1)^{2}(z+1)=z^{5}-z^{4}-z^{3}+z^{2}$

Graph of the polynomial y = x^3 + 2*x^2 - 7*x + 4 with its roots (zeros) -4 and 1 marked.

The polynomial

  $f(x)=(x+4)(x-1)^{2}=x^{3}+2x^{2}-7x+4$

has roots:

1 of multiplicity 2
-4 of multiplicity 1

rboyce1000

p(z) = z^4 + O(z^2), where the four roots of p(z) are:

one fixed at the origin,
the remaining three forming the vertices of an equilateral triangle centered at the origin and rotating.

create polynomial with desired properities

- f(z) = z*g(z) with root at origin
- g(z) is a 3-rd root of unity = $x^{2}+x+1$

f(z) = z(z^2+z+1)

One can check it with Maxima CAS

(%i1) solve([z*(z^2+z+1)=0],[z]);
                      sqrt(3) %i + 1      sqrt(3) %i - 1
(%o1)          [z = - --------------, z = --------------, z = 0]
                            2                   2
(%i2)

to rotate it around origin let's change 1 with : $\quad e^{2\pi it}$ wher t is a proper fraction in turns

f_{t}(z)=z(z^{2}+z+e^{2\pi it})

See : Vector_field short videos by boyce1000

one parameter

System of 2 equations:

${\begin{cases}f_{c}^{p}(z_{p})=z\\{\frac {d}{dz}}f_{c}^{p}(z_{p})=r*e^{2\pi \theta i}\end{cases}}$

where:

$f_{c}$ is a rational function with one parameter c
$\ f_{c}^{(p)}(z)$ is the $p$ -fold composition of $f_{c}\,$ with itself
$z_{p}$ is a cyclic point ( point of limit cycle)
p is a period of the cycle
$\lambda =r*e^{2\pi \theta i}$ is a multiplier^[10] ( complex number)
$r=|\lambda |$ is a stability of the cycle ( Real number )

Input :

function $f$
p ( integer)
r ( real number)
$\theta$ ( real number or rational number)

Unknowns ( solutions or output):

parameter c ( complex number)
periodic point $z_{p}$ ( complex number)

Maxima CAS program:

/*

batch("m.mac");

*/

display2d:false$
kill(all)$
ratprint:false$

/* complex quadratic polynomial */
f(z,c):= z*z+c $

/* iterated function */
F(z, c, n) :=
       if n=1 then f(z,c)
        else f(F(z, c, n-1),c)$
        
        
        
/*  multiplier = first deric=vative */        
m(z,c,p):= diff(F(z,c,p),z,1)$

l(r,t) := float(rectform(r*exp(2*%pi*t*%i)))$

/* input */

p:5$
r:1.0$
t:0$

/* system of equations */
e1: F(z,c,p)=z;
e2: m(z,c,p)=l(r,t);

/* 
output = solutions = 2 complex number: c, z 
*/

s:solve([e1,e2])$
s:map('float,s)$
s:map('rectform,s);

Example output:

For :

p = 3
r=1.0
t=0.0

[
[z = 0.5,c = 0.25],
[z = (-0.4330127018922193*%i)-0.25,c = (-0.6495190528383289*%i)-0.125],
[z = 0.4330127018922193*%i-0.25,c = 0.6495190528383289*%i-0.125],
[z = -0.05495813133539004,c = -1.75],
[z = 1.301937809824245,c = -1.75],
[z = -1.746979634104245,c = -1.75]
]

For :

p = 5
r=1.0
t=0.0

[
[z = 0.5,c = 0.25],
[z = 0.4755282581475767*%i+0.1545084971874737,c = 0.3285819450744551*%i+0.3567627457812099],
[z = 0.1545084971874737-0.4755282581475767*%i, c = 0.3567627457812106-0.3285819450744586*%i],
[z = 0.2938926261462365*%i-0.4045084971874737, c = 0.5316567552200239*%i-0.4817627457812153],
[z = (-0.2938926261462365*%i)-0.4045084971874737, c = (-0.5316567552199369*%i)-0.481762745781224],
[z = -0.003102011282477321,c = -1.985409652076318],
[z = 0.0109289978340113,c = -1.860587002096436],
[z = 8.008393221517376E-4*%i-0.01213161194929343, c = 1.100298437397382*%i-0.1978729466687337],
[z = (-8.008393221517376E-4*%i)-0.01213161194929343, c = (-1.100298437397305*%i)-0.1978729466687667],
        [z = 0.02151217276434695*%i-0.005267866463337371, c = 0.3797412022535638*%i-1.256801993945385],
        [z = (-0.02151217276434695*%i)-0.005267866463337371, c = (-0.3797412022517599*%i)-1.256801993944077],
        [z = 0.02591758988716001*%i+0.0096648625988135, c = 0.9868115621249533*%i-0.04506136597934137],
        [z = 0.0096648625988135-0.02591758988716001*%i, c = (-0.9868115621250132*%i)-0.04506136597930513],
        [z = -0.02506558296814108,c = -1.624396967608546],
        [z = 0.02532354987824971*%i-0.0286751769590709, c = 0.6415066667139064*%i+0.3599331333357185],
        [z = (-0.02532354987824971*%i)-0.0286751769590709,
         c = 0.3599331333357186-0.6415066667139071*%i], [z = 0.7018214526647177,c = -1.860587002096436],
        [z = 0.5745382937725365*%i+0.1798116252110209, c = (-0.379741202251533*%i)-1.25680199394442],
        [z = 0.1798116252110209-0.5745382937725365*%i, c = 0.3797412022514344*%i-1.256801993944486],
        [z = -0.5997918293000261,c = -1.624396967608546],
        [z = 0.6400543521659254*%i+0.3601141169309163, c = 0.6415066667138928*%i+0.3599331333356947],
        [z = 0.3601141169309163-0.6400543521659254*%i, c = 0.3599331333356951-0.6415066667138929*%i],
        [z = 0.747361547631752*%i+0.4122389750905872, c = 0.3599331333377524-0.6415066667118048*%i],
        [z = 0.4122389750905872-0.747361547631752*%i,c = 0.6415066667118131*%i+0.3599331333377574],
        [z = -1.264646754738656,c = -1.624396967608546],
        [z = 0.838427461519175*%i+0.1867295812979602,c = (-0.9868115621248*%i)-0.04506136597962632],
        [z = 0.1867295812979602-0.838427461519175*%i, c = 0.9868115621248269*%i-0.04506136597961512],
        [z = 1.012227741688957,c = -1.624396967608546],
        [z = 0.6736931444481549*%i-0.7131540376767388,  c = 0.9868115621009495*%i-0.04506136566593825],
        [z = (-0.6736931444481549*%i)-0.7131540376767388, c = (-0.9868115621015654*%i)-0.04506136566602404],
        [z = 0.6816651712455555*%i+0.8064792250322852,  c = (-1.100298438532418*%i)-0.1978729463920518],
        [z = 0.8064792250322852-0.6816651712455555*%i,c = 1.100298438531886*%i-0.197872946387467],
        [z = 0.9873125420152975*%i-0.04563967787575593, c = 0.9868115621249436*%i-0.04506136597927069],
        [z = (-0.9873125420152975*%i)-0.04563967787575593, c = (-0.9868115621249249*%i)-0.04506136597929692],
        [z = -1.368033648790746,c = -1.860587002096436],
        [z = -1.623768668573244,c = -1.624396967608546],
        [z = 1.600752508361204,c = -1.860587002096436],
        [z = 0.8177857184842046*%i-0.8491638964763748,  c = 0.6415066726649287*%i+0.3599331357137042],
        [z = (-0.8177857184842046*%i)-0.8491638964763748, c = 0.3599331357115682-0.6415066726792946*%i],
        [z = -1.860467532467532,c = -1.860586580956207], 
        [z = 0.1585230889211015*%i+1.129895436404861,  c = (-0.3797412017812437*%i)-1.256801993890818],
        [z = 1.129895436404861-0.1585230889211015*%i,  c = 0.3797412020742688*%i-1.256801993924219],
        [z = 1.102491882350288*%i+0.07994573682221373, c = 0.641506666713125*%i+0.3599331333375105],
        [z = 0.07994573682221373-1.102491882350288*%i, c = 0.3599331333375118-0.641506666713142*%i],
        [z = 1.10027900645412*%i-0.1977264120044163,c = 1.100298437399976*%i-0.1978729466589521],
        [z = (-1.10027900645412*%i)-0.1977264120044163, c = (-1.100298437392994*%i)-0.1978729466579122],
        [z = 0.3795145554958574*%i-1.257237017109811,  c = 0.3797412012322979*%i-1.256801993538778],
        [z = (-0.3795145554958574*%i)-1.257237017109811, c = (-0.3797412011893692*%i)-1.256801993401957],
        [z = 0.8966903093631682*%i-1.01776444141452, c = 0.986811439368143*%i-0.04506141337632084],
        [z = (-0.8966903093631682*%i)-1.01776444141452, c = (-0.9868114393633113*%i)-0.04506141338736716],
        [z = 1.407944514501891,c = -1.985409652076318],
        [z = 0.7215120925377011*%i+1.234881318742427,  c = (-1.100298500720014*%i)-0.1978727350763138],
        [z = 1.234881318742427-0.7215120925377011*%i,c = 1.100298500782114*%i-0.1978727352231734],
        [z = 0.6651899971189704*%i-1.369391104706556, c = 1.100298438532065*%i-0.1978727774731155],
        [z = (-0.6651899971189704*%i)-1.369391104706556, c = (-1.100298478086625*%i)-0.1978727911942495],
        [z = 0.1731238730127708*%i-1.554564024233688,  c = 0.3797412149717089*%i-1.256801976456581],
        [z = (-0.1731238730127708*%i)-1.554564024233688, c = (-0.3797411926534995*%i)-1.256801968631482],
        [z = 1.842105908761944,c = -1.985410334346504],
        [z = 1.956403762662807,c = -1.985409652076318]
        ]

Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley

Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley by izaytsev0

main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part
2 windows
- left: limbs from period 2 component
- right: limbs from period 1 componnet
in each window one can see limb p/q from 14/30 on the right and increasing p ?

Compare with Real dense fractal Zoom! Part 2 by SeryZone Arts

References

[1] ractalforums: julia-sets-true-shape-and-escape-time

[2] ractalforums : constructing-polynomials-whose-julia-set-resemble-a-desired-shape

[3] ractalforums : constructing-polynomials-with-attracting-cycles

[4] The W. Thurston algorithm applied to real polynomial maps Araceli Bonifant, J. Milnor, S. Sutherland Published 15 May 2020

[5] The fundamental theorem of algebra in wikipedia

[6] Ed Pegg Jr "The Fundamental Theorem of Algebra" http://demonstrations.wolfram.com/TheFundamentalTheoremOfAlgebra/ Wolfram Demonstrations Project Published: November 10 2011

[7] Factor theorem in wikipedia

[8] Multiplicity_of_a_root_of_a_polynomial in wikipedia

[9] ractalforums.org : julia-sets-true-shape-and-escape-time

[10] Stability_of_periodic_points_(orbit)_-_multiplier in wikipedia

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Fractals/Iterations in the complex plane/construct

Contents

Shape

Modelling or shaping

Conversions and evolutions

Examples

roots

rboyce1000

one parameter

Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley

See also

References